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Home Quiz 1 MATH 251, Section 505
Due, February, 25th, 2016
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first name : . . . . . . . . . . . . . . . . . .
”An Aggie does not lie, cheat or steal, or tolerate those who do”
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Write up your result, detail your calculations if necessary and BOX your final answer.
Your final work have to be neat, so use pencil first if you want.
1. [6pts] Find the equation of the plane passing through A(1, 1, −3), B(−2, 4, 1) and C(0, 2, 1).
2. [8pts] Determine whether the planes P1 : 2x + y = 1 − z and P2 : 2x − 3z = −4y + 5 are parallel,
orthogonal, or neither. Find cosine of the angle between the planes.
3. [10pts] Find the domain of z = f (x, y) =
z ≥ 0?
4. [2pts] What is the domain of f (x, y) =
p
3 − 2x2 − 2y 2 and sketch it. What is the range of f when
√
x + ln(y) −
√x .
y
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5. [6pts]Let C be the curve with equations r(t) = h2t2 , 2t − 2, ln(3t + 1)i. Find an equation of the tangent line to C at (2, 0, 2 ln 2).
6. [8pts] Identify the level curves of the surface given by z = 4e−(2x
2 +2y 2 )
.
7. [10pts] Compute the second partial derivatives of f (x, y) = y 2 x + 5y 2 e−x − cos(x2 ).
8. [10pts] Use differentials to approximate ln(2.03 − (0.99)2 ).
9. Given the surface x2 y 3 z = 18 and the point P (3, 1, 2).
(a) [5pts] Find an equation of the tangent plane to the given surface at P .
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(b) [8pts] Find the equation of the normal line at P to the given surface.
x
+ 2ez .
y
(a) [5pts] Find the direction in which f increases most rapidly at the point (1, 2, 0).
10. Let f (x, y, z) = y 3 +
(b) [5pts] Find the directional derivative of f at the point (1, 2, 0) in the direction < 4, 0, −1 >.
11. Let f (x, y) = xy(2 − x − y).
(a) [10pts] Find all critical points of f .
(b) [10pts] Classify the critical points (local maximum, local minimum or saddle points).
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